Ignore Limit is the whole problem. Here is how to take the button away.

The grey screen is up, and one of the buttons on it says Ignore Limit.

You set that limit. You meant it at the time. And the phone, which has no way of telling that the person reading this sentence is the same person who set it, is politely offering to cancel the whole arrangement for you with one thumb and no questions.

The button comes off. Not by deleting it, because you cannot delete it, but by replacing it with one that wants four digits first. That takes two settings and about ninety seconds. Here is the whole procedure before any of the explaining.

1. Set a Screen Time passcode. Settings › Screen Time, scroll to the bottom, Lock Screen Time Settings. Four digits, then the same four again. On iOS 15 and earlier that row reads Use Screen Time Passcode. Forty seconds.
2. Decide about the recovery Apple Account on purpose. iOS offers to attach one so the passcode can be reset if you forget it. Saying yes is sensible and it is also a second way in. Say yes or no deliberately rather than by reflex, then go and read the piece about where the digits live later.
3. Open your limit and turn on Block at End of Limit. Settings › Screen Time › App Limits, tap the limit, and the switch is now at the bottom of that screen. It was not there ten minutes ago. Twenty seconds.
4. Do it again inside every other limit you have. The switch belongs to the limit, not to Screen Time, and setting the passcode does not go back and flip any of them.
5. If you use Downtime, its switch is called Block at Downtime. Settings › Screen Time › Downtime. Same hiding place, same rule: no passcode, no switch.

That is the article. Everything below is why each of those steps was hard to find, and the one thing this fixes and the one thing it does not.

The toggle is not missing. It has not been drawn yet.

Search for why Block at End of Limit is missing and you find hundreds of people describing exactly the screen you were looking at: App Limits, a limit, a list of apps, a schedule, and nothing at the bottom. No switch. Nothing is broken and you did not fail to scroll.

iOS only draws that switch once a Screen Time passcode exists on the device. It never says so. Not on that screen, not on the App Limits screen above it, and not in Apple's own support page for setting up Screen Time, which walks you through the passcode and never mentions the setting the passcode unlocks.

You did not fail to find the setting. It was not on the screen.

The reasoning behind hiding it is defensible. A hard block with no passcode behind it would shut you out of an app until midnight with nothing to type and no way back in, and an operating system that can lock a person out of their own banking app with no key is an operating system that gets a phone call from a lawyer. So Apple waits until you have cut a key before showing you the door that uses it.

The cost of that decision is that the single most useful setting for an adult who is losing to their own phone is invisible to every adult who has not already guessed at a different setting first.

A passcode on its own does nothing to the button

This is the part that catches the people who already did the sensible thing. Setting a Screen Time passcode locks the settings so nobody can edit your limits without the digits. That is all it does. It does not touch the shield.

With a passcode set and Block at End of Limit still off, the grey screen carries on saying Ignore Limit, and tapping it still hands you the rest of the day without asking you for anything at all. The lock is on the cupboard. The door is still open.

So if you set the passcode last month, felt organised about it, and then found yourself in Instagram at nine forty anyway, you were not imagining it and the passcode was not broken. It was never in the path.

The switch belongs to the limit, so old limits keep the old behaviour

If you built four limits over the past few months, you now have four separate screens to visit. Block at End of Limit lives inside each individual limit, underneath the apps and the schedule, and not on the App Limits list where you would look for something that applied to all of them.

Twenty seconds each. The one people forget is always the first one they ever made, months ago, for the app they have mostly stopped worrying about, which is also the app they will open tonight.

What the grey screen says once you have done it

The button stops saying Ignore Limit. It says Ask For More Time.

Tap it and you still get one minute, free, no digits. That minute is deliberate and it is worth keeping. It exists so you can finish sending the message you were halfway through when the limit landed, which is a genuinely humane thing for an operating system to offer.

Anything past that minute wants the passcode first. Fifteen more minutes, an hour, the rest of the day: all of them, four digits, every time.

So it is not a wall. It is a toll booth. But the toll has to be paid on purpose, and a deliberate act at nine forty is a completely different animal from a thumb landing on a button it has landed on eleven times before.

Downtime has the same switch under a different name

If your rule is a bedtime rather than a budget, App Limits is the sideshow and Downtime is the main event. It has its own version of this, called Block at Downtime, at Settings › Screen Time › Downtime, and it hides in exactly the same way for exactly the same reason.

Left off, the Downtime shield offers to ignore itself. Turned on, it asks for the passcode like the other one. Most people who set a bedtime never see the switch, conclude Downtime does not really do anything, and quietly stop using it.

The three switches, on one page

Now the part about the key you cut for yourself

Do all of the above. It is ninety seconds and it removes the most-pressed escape hatch on the phone, and if you stop reading here you have got what you came for.

Then be honest about what you have built. You chose the passcode. It is four digits, and unless you did something unusual, they are four digits your thumb already knows from somewhere. The shield now asks a question that you happen to be carrying the answer to.

That is not nothing. A question at nine forty is a great deal more than a button at nine forty, and plenty of evenings end right there, at the keypad, with the phone going face down. But some evenings do not, and on those evenings the difference between a lock and a delay becomes very obvious very quickly.

Which is why the durable answers all involve a second party. Someone who was not in the room at nine, whose mind does not change because you are tired. That can be an actual person: hand your partner the four digits and ask them to type them, and it works extremely well and costs you something socially every single time, which is precisely why it works.

Or it can be something that behaves like one. We make one of those, so here is the part where we say so. Scrappy is a small dog who stands in front of the apps you name. He does not know the Screen Time passcode and does not want it, because a passcode you can supply on demand is not a lock, it is a formality.

Getting past him costs a treat. Treats come from walking, roughly two thousand steps for one, and a treat buys five honest minutes. They are never sold. They never will be, no matter how the year is going, because the entire point is that the currency is something you cannot manufacture from the sofa at nine forty.

He also gets less patient the more you knock. First knock he is polite about it. By the fourth he is one word, set very large, and it is not a word anyone argues with. Apps you are properly serious about can be set to firm, which makes you sit for ten seconds before he will even consider the treat, or absolute, where he will not take it at all.

He knows three rules and no fourth: quiet hours, a daily limit, an open limit. There is a removal lock for the delete-and-reinstall route, and switching that off takes a day, which is deliberately longer than an urge lasts. There is one emergency pass a month, for boarding passes and real emergencies, and he does not scold you for using it. No account, no sign-up, nothing on a server of ours: your Screen Time numbers are drawn by a sandboxed Apple extension that cannot phone home.